I'm posting this as a reference for myself. Yes, there are other answers to this problem but this is a bit different.
Instead of booting from liveDVD as other answers suggest, you may display GRUB menu at boot (e.g. by pressing and holding the Shift key) and then choose 'Advanced options for Ubuntu' in Ubuntu 14 LTS (in Ubuntu 12 LTS it is 'Previous linux versions'). Confirm the uppermost/latest version to boot. When OS is up and running, you may perform the following procedure.
sudo mount /dev/sda1 /mnt
sudo mount --bind /dev /mnt/dev
sudo mount --bind /proc /mnt/proc
sudo mount --bind /sys /mnt/sys
sudo chroot /mnt
Note that you should verify that
/dev/sda1 is really your boot disk.
Procedure was suggested here.
Is anybody aware of potential problems due to the fact that my solution uses the older version of the OS? By now, system where this was applied, works fine.